Let's analyze the mathematical question.

The problem asks which of the given options are signified by the expression:

$\lim_{x \to a} f(x) = \infty$

Key Concept:

• This notation means that as $x$ approaches $a$, the function $f(x)$ increases without bound. In other words, $f(x)$ grows larger and larger indefinitely.
• However, this does not mean that $f(a)$ itself is infinite or even defined. The limit only describes the behavior of $f(x)$ as $x$ approaches $a$; it does not make any guarantees about the value of $f(x)$ at $x = a$.

Analyzing Each Option:

1. "The limit exists and equals infinity."

   • Incorrect. The phrase "the limit exists" refers to a finite value, but $\infty$ is not a finite number. When the limit grows without bound, it is more precise to say the limit diverges to infinity rather than that it "exists."

2. "This means that $f(a) = \infty$."

   • Incorrect. The limit does not tell us anything about the value of $f(x)$ at $x = a$. In fact, $f(a)$ may not even be defined.

3. "The value of $f(x)$ becomes infinite when $x$ approaches $a$."

   • Correct. This accurately describes the behavior of the function as $x$ gets arbitrarily close to $a$.

4. "The value of $x$ goes out to infinity, when it gets close to $a$."

   • Incorrect. This statement misunderstands the relationship between $x$ and $f(x)$. It is $f(x)$ (the output), not $x$ (the input), that increases without bound as $x$ approaches $a$.

5. "As $x$ approaches $a$, the value of $f(x)$ increases without bound."

   • Correct. This is another way of correctly describing the behavior indicated by $\lim_{x \to a} f(x) = \infty$.

Final Answer:

The correct options are:

• 3: "The value of $f(x)$ becomes infinite when $x$ approaches $a$."
• 5: "As $x$ approaches $a$, the value of $f(x)$ increases without bound."

Would you like a deeper explanation of limits or related examples?

5 Related Questions:

1. What is the difference between a finite limit and an infinite limit?
2. How do you evaluate $\lim_{x \to a} f(x)$ when $f(x)$ involves a rational function?
3. Can $\lim_{x \to a} f(x) = \infty$ occur if $f(x)$ is discontinuous at $x = a$?
4. What happens if $\lim_{x \to a} f(x) = -\infty$? How does it differ from $\infty$?
5. How do horizontal and vertical asymptotes relate to limits?

One Tip:

When analyzing limits involving $\infty$, always carefully distinguish between the behavior of the function near the point and the actual value of the function at the point.